Relativistic Bernstein modes are not totally undamped, but have a small, negative definite imaginary frequency component which peaks where the frequency is closest to the rest cyclotron harmonic

Diver, D.A.

Laing, E.W.

English

Enlighten

      Laing, E W and Diver, D A (2005) On damped Bernstein modes in a weakly relativistic pair plasma. Physical Review E 72(3):036409.       http://eprints.gla.ac.uk/3154/     Glasgow ePrints Service http://eprints.gla.ac.uk On damped Bernstein modes in a weakly relativistic pair plasmaE W Laing and D A DiverDept of Physics and Astronomy, Kelvin Building,University of Glasgow, Glasgow G12 8QQ, Scotland UK.∗(Dated: July 12, 2005)Relativistic Bernstein modes are not totally undamped, but have a small, negative definite imagi-nary frequency component which peaks where the frequency is closest to the rest cyclotron harmonic.PACS numbers: 52.25.Dg, 52.25.Xz, 52.27.Ep, 52.27.Ny, 52.35.FpI. INTRODUCTIONIn our previous paper on Bernstein modes in a weaklyrelativistic treatment based on Maxwellian distributions[1] we claimed that contributions due to the singularitieswhich occurred at the resonances γ2 = n2/ωˆ2 exactlycancelled, leaving the modes undamped. However it hassince been pointed out [2] that the path of integrationproposed in [1] to avoid the singularities was incorrect.The correct path should have been below the singularityon the negative pˆ‖-axis, and above it on the positive pˆ‖-axis. As a result, the contributions add, and the modesare damped.In this brief article we calculate the damping of therelativistic Bernstein modes, assuming that the damp-ing is weak. More precisely, given that the frequencyof a Bernstein wave of normalised wave-number κ =(2/a)1/2k⊥c/Ω0 is ωˆ = ω/Ω0, we now write ωˆ = ωˆr+ iωˆi,and assume that |ωˆi/ωˆr|  1. We will use the same no-tation throughout as in [1], so that a = mec2/(kBT ) isthe relativistic parameter, and Ω0 = eB0/me is the restcyclotron frequency for an electron or positron.The full dispersion relation obtained in [1] isωˆ2 − 2ωˆ2p =4ωˆ2pa5/2√2πωˆ2kˆ2⊥∞∑n=1n4∞∫0dpˆ⊥pˆ⊥J2n(kˆ⊥pˆ⊥)e−apˆ2⊥/2×∞∫−∞dpˆ‖e−apˆ2‖/21 + pˆ2‖ + pˆ2⊥ − n2/ωˆ2(1)where ωˆp = ωp/Ω0 is the normalised plasma frequency,pˆ‖ = p‖/(mec) and pˆ⊥ = p⊥/(mec).Consider the properties of the second integral inEq. (1). In the limit |ωˆi/ωˆr|  1,n2ωˆ2≈ n2ωˆ2r(1− 2i ωˆiωˆr)(2)∗enquiries to d.diver@physics.gla.ac.uk.v - planeΓ v0-v0.real axisFIG. 1: Definition of the contour for the integral in Eq. (9).Hence the integral becomesI =∞∫−∞dpˆ‖e−apˆ2‖/21 + pˆ2‖ + pˆ2⊥ − n2/ωˆ2r + 2in2ωˆi/ωˆ3r(3)For convenience, deﬁneb2n = n2/ωˆ2r − 1 (4)so that it is apparent that a diﬀerent treatment of theintegral I is required depending on the sign of b2n, andthe relative sizes of b2n and pˆ2⊥.We will use throughout the notationx2n = ab2n/2 (5)yn = an2ωˆi/ωˆ3r < 0 (6)v2 = apˆ2‖/2 (7)x2 = x2n − apˆ2⊥/2 (8)Case (i) b2n > 0, b2n > pˆ2⊥The integral terms in Eq. (1) can be written in thesimpler formI =√2/axn∫0xe−x2n+x2J2n(κ√x2n − x2) dx×∫Γe−v2dvv2 − x2 + iyn (9)in which Γ is the Landau contour given in Fig. 1. Thepoles are at ±v0, where v0 =√x2 − iyn.For the singular integral in (9), the contribution fromthe poles is given by−2πie−x2+iynv0(10)using the Residue theorem. The integration along thereal line then yields∞∫−∞(v2 − x2)e−v2dv(v2 − x2)2 + y2n− iyn∞∫−∞e−v2dv(v2 − x2)2 + y2n(11)with all other contributions cancelling. Note that theintegrals above are both real.Case (ii) b2n > 0, b2n < pˆ2⊥Here the pˆ‖ integral takes the form∞∫−∞e−v2dvv2 + x2 + iyn(12)and is non-singular. This integral can be expressed interms of real and imaginary parts as∞∫−∞e−v2(v2 + x2)(v2 + x2)2 + y2ndv− iyn∞∫−∞e−v2(v2 + x2)2 + y2ndv (13)When yn = 0, the second integral vanishes, as expected.Case (iii) n < ωˆrThis is similar to case (ii), but now b2n < 0; the pˆ⊥integral is as before, subject to this proviso.The dispersion relation for frequency m− 1 < ωˆr < mwhere m ≥ 2 is an integer and C = 8√πωˆ2pa/κ2 can bewrittenωˆ2r + 2iωˆiωˆr − 2ωˆ2p =Cωˆ2r(1− 2i ωˆiωˆr)×[m−1∑n=1n4AIIIn +∞∑n=mn4(AIn + AIIn − iBn)](14)in whichAIn = e−x2nxn∫0xex2J2n(κ√x2n − x2)×[q1(x, yn)− iynq2(x, yn)] dx (15)AIIn = e−x2n∞∫0xe−x2J2n(κ√x2n + x2)×[p1(x, yn)− iynp2(x, yn)] dx (16)AIIIn = e−x2n∞∫|xn|xe−x2J2n(κ√x2 + x2n)×[p1(x, yn)− iynp2(x, yn)] dx (17)Bn = 2e−x2nxn∫0xJ2n(κ√x2n − x2)√x2 − iyneiyndx (18)and where the following additional function deﬁnitionshave been used:p1(x, y) =1π∞∫−∞dve−v2(v2 + x2)(v2 + x2)2 + y2(19)p2(x, y) =1π∞∫−∞dve−v2(v2 + x2)2 + y2(20)q1(x, y) =1π∞∫−∞dve−v2(v2 − x2)(v2 − x2)2 + y2 (21)q2(x, y) =1π∞∫−∞dve−v2(v2 − x2)2 + y2 (22)Note that in the limit yn → 0, p1 → ex2erfc(x), q1 →ie−x2erf(ix) and AIIn and AIIIn become purely real; thesethen agree with the deﬁnitions given in Eqs. (48-49) of[1]. The real part of AIn similarly agrees with Eq. (47)of [1], but there is an imaginary contribution from thechanged contour.The procedure now, as discussed earlier, is to extractthe real and imaginary parts of the dispersion relation,and to make the assumption that the real part is rela-tively unchanged from our earlier paper [1]. In this way,we can use the imaginary part of the dispersion relationto deﬁne an implicit relationship that will yield ωˆi as afunction of ωˆr and the plasma parameters. It will turnout that we must retain the eﬀects of yn beyond the linearapproximation in order to avoid singular integrals. Thisis not the case when considering the real part, hence jus-tifying the approximation of setting ωˆi = 0 there.In order to separate the real and imaginary parts of thedispersion relation, we introduce the following notationfor AIn,AIn = AIn1 − iynAIn2 (23)AIn1 = e−x2nxn∫0xex2J2n(κ√x2n − x2)q1(x, yn)dx (24)AIn2 = e−x2nxn∫0xex2J2n(κ√x2n − x2)q2(x, yn)dx (25)with the obvious extension to AIIn and AIIIn .For Bn, we note that(x2 − iyn)−1/2 =√rn + x2 − i√rn − x2√2rn(26)r2n = x4 + y2n (27)yn < 0 (28)and so we can writeBn = Bn1 + iBn2 (29)Bn1 =√2e−x2nxn∫0dxxrnJ2n(κ√x2n − x2)×[√rn + x2 cos yn +√rn − x2 sin yn](30)Bn2 =√2e−x2nxn∫0dxxrnJ2n(κ√x2n − x2)×[√rn + x2 sin yn −√rn − x2 cos yn](31)With this notation we can collect the imaginary terms inthe dispersion relation to get2iωˆrωˆi = −i Cωˆ2r[m−1∑n=1n4ynAIIIn2+∞∑n=mn4yn(AIn2 + AIIn2)+∞∑n=mn4Bn1]−i2Cωˆiωˆ3r[m−1∑n=1n4AIIIn1+∞∑n=mn4(AIn1 + AIIn1)+∞∑n=mn4Bn2](32)Now since we have assumed that |ωˆi/ωˆr|  1, and there-fore that the real part of the dispersion relation can becalculated by ignoring ωˆi contributions, we haveωˆ2r−2ωˆ2p =Cωˆ2r[m−1∑n=1n4AIIIn1+∞∑n=mn4(AIn1 + AIIn1 + Bn2) ](33) 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 0.5  1  1.5  2  2.5  3  3.5  4  4.5 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1FIG. 2: Pluses (+) show the upper loop of the dispersioncurve ωˆr (left y-axis) versus κ (x-axis) for a = 10, and 1 <ωˆr < 2. The crosses (×) show how −ωˆi (right y-axis) variesas a function of κ along the dispersion curve.This allows us to simplify the expression for ωˆi:ωˆi =−C∞∑n=mn4Bn14ωˆr(ωˆ2r − ωˆ2p) +aCωˆ3r[m−1∑n=1n6AIIIn2 +∞∑n=mn6(AIn2 + AIIn2) ](34)Note that Eq. (34) deﬁnes ωˆi implicitly, since ωˆi appearsin functional dependencies on the right hand side. Tosolve Eq. (34) for ωˆi requires an iterative scheme, con-verging on the true value. The results of such a schemeapplied to the cases of a = 10 and a = 20, for frequenciesin the range 1 < ωˆr < 2 and 2 < ωˆr < 3 and for ωˆp = 3are shown in the ﬁgures. As in [1], harmonics above thesixth order could be neglected for these cases.It is possible to derive a very simple approximationfor the right-hand side of Eq. (34). Noting that AIm2 isdominant among the terms in the denominator (by virtueof its dependence on the function q2) , and similarly Bm1is the largest contributor to the numerator, it is possibleto writeωˆi ≈ − ωˆ3rBm1am2AIm2(35)if we retain only these dominant terms, and if we neglectthe term 4ωˆr(ωˆ2r− ωˆ2p) compared with aCn6AIm2/ωˆ3r . Us-ing the deﬁnition of yn from Eq. (6), we have eventuallyym ≈ −Bm1AIm2(36)which generally overestimates the value of |ωˆi| by up to20% when compared with the result of retaining all therelevant summation terms. 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1  1.5  2  2.5  3  3.5  4  4.5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08FIG. 3: Caption as for Fig. 2, showing the real and imaginaryparts on the lower loop of the same dispersion curve. 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9 1.92 0.5  1  1.5  2  2.5  3  3.5  4  4.5 0.03 0.035 0.04 0.045 0.05 0.055FIG. 4: Pluses (+) show the upper loop of the dispersioncurve ωˆr (left y-axis) versus κ (x-axis) for a = 20, and 1 <ωˆr < 2. The crosses (×) show how −ωˆi (right y-axis) variesas a function of κ along the dispersion curve.II. SUMMARYThis short article updates our original calculations[1] by showing that Bernstein modes in relativistic pairplasmas are indeed damped, albeit very weakly. Theclassical electron-ion picture suggests that this dampingshould occur close to the classical cyclotron resonancesat ω ≈ nωc [3]. Our computed damping terms are largestclosest to the cyclotron harmonics on the upper loop ofeach dispersion curve; ωˆi is comparatively smaller on thelower branch mainly because xm is larger in that lowerfrequency part of the curve, and so AIm2 is more domi-nant than Bm1, leading to a smaller ωˆi. This can readilybe seen from the simple approximation given in Eq. (36).Note also that we have shown in earlier work [1] thatrelativistic eﬀects signiﬁcantly change the shape and lo-cation of the dispersion curves for pair-plasma Bernsteinmodes such that they do not remain localised near a par-ticular harmonic. In fact, most of the Bernstein modedispersion curves are free from signiﬁcant damping, andtherefore the associated waves propagate largely without 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1  1.5  2  2.5  3  3.5  4  4.5 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045FIG. 5: Caption as for Fig. 4, showing the real and imaginaryparts on the lower loop of the same dispersion curve. 2.55 2.6 2.65 2.7 2.75 2.8 2.85 2.9 1  2  3  4  5  6  7  8 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075FIG. 6: Pluses (+) show the upper loop of the dispersioncurve ωˆr (left y-axis) versus κ (x-axis) for a = 20, and 2 <ωˆr < 3. The crosses (×) show how −ωˆi (right y-axis) variesas a function of κ along the dispersion curve.hindrance.III. AKNOWLEDGMENTSIt is a pleasure to acknowledge valuable and construc-tive comment from R A Cairns and C N Lashmore-Davies. We are grateful to PPARC for partial funding ofthis research.[1] D. A. Keston, E. W. Laing and D. A. Diver, Phys. Rev.E 67, 036403 (2003)[2] R. A. Cairns and C. N. Lashmore-Davies, private commu-nication 2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65 2.5  3  3.5  4  4.5  5  5.5  6  6.5 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045FIG. 7: Caption as for Fig. 6, showing the real and imaginaryparts on the lower loop of the same dispersion curve.[3] R. A. Cairns and C. N. Lashmore-Davies, p.430, CP595Radio Frequency Power in Plasmas: 1414 Topical Conf.,T. K. Mau, J. deGrassie, editors, AIP 0-7354-0038-5(2001)

On damped Bernstein modes in a weakly relativistic pair plasma

Enlighten: Publications

http://eprints.gla.ac.uk/3154/1/berndamp224fin.pdf

On damped Bernstein modes in a weakly relativistic pair plasma

Abstract

Similar works

Full text

Available Versions

Enlighten

Enlighten: Publications